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This is a crosspost of this MSE question.

A topological space is connected if it's not the coproduct of two non-trivial spaces. Equivalently, it is connected if the copresheaf it represents preserves coproducts.

A topological space is irreducible if it's not the union of proper closed subsets, or alternatively if every pair of non-trivial opens has inhabited intersection. Does irreducibility admit a functorial description?

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  • $\begingroup$ Maybe in terms of the corresponding locale? Just a thought... $\endgroup$
    – David Roberts ♦
    Commented Aug 16, 2017 at 22:50
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    $\begingroup$ By the way, I wish I could +1 your user page, handy code and all. $\endgroup$
    – David Roberts ♦
    Commented Aug 16, 2017 at 22:54
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    $\begingroup$ Martin Brandenburg points out that there's a "brute force" categorical characterization of the Sierpinski space. From there, you can talk about closed sets categorically and so also irreducibility. But somehow this is not terribly satisfying. $\endgroup$
    – Tim Campion
    Commented Aug 17, 2017 at 2:54
  • $\begingroup$ You could try to find a similar categorical description of the space $T$ with elements $\{x,y,z\}$ and nontrivial closed sets $\{x,y\}$, $\{y,z\}$, and $\{y\}$. Then maps $f:X\to T$ with $x$ and $z$ in the image indicate that $f^{-1}(\{x,y\})$ and $f^{-1}(\{y,z\})$ are proper closed subsets covering $X$. $\endgroup$
    – MTyson
    Commented Aug 17, 2017 at 18:37

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